Prediction of flow of fluids through a reservoir has been used for certain scenarios (e.g., well placement, production optimization, etc.) where it is desirable for supporting business/investment decisions. For efficient recovery of oil and gas from a reservoir, a good understanding of the subsurface attributes and its constituents is vital.
Conventionally, production data consisting of measurements of pressures in the wells, along with liquid (oil and water) and gas flow rates, is used in attempt to recover the subsurface attributes. The process in which this is performed is called history matching. In this process, the model parameters (such as permeability, porosity, skin, seal factors) are altered so that simulation of flow would match the recorded production data at the wells. There are several strategies for updating the model parameters, including manual trial and error. The most widely accepted approach is based upon non-linear optimization. In such non-linear optimization, the problem is cast as minimization of an objective function that consists of a measure of misfit (likelihood) between the real measured data and the one that is simulated for a choice of model parameters. e.g.:
            m      ^        =                                        arg            ⁢                                                  ⁢            min                    m                ⁢                                  ⁢        J            ≡                                                  D              ⁡                              (                                                      P                    ⁡                                          (                                              u                        ⁡                                                  (                                                      m                            ;                            y                                                    )                                                                    )                                                        ,                                      d                    ⁡                                          (                      y                      )                                                                      )                                      ︸                                data            ⁢                                                  ⁢            misfit                          +                                            S              ⁡                              (                m                )                                      ︸                    regularization                                s      .      t      .                          ⁢              g        (                  m          ;          u                )              =    0    constraintswhere m denotes the model parameters, J is the objective function, D is a noise model, P is a function that converts the state u (saturation and pressure for flow in porous medium) into simulated measurement, y denotes the experimental design setup and d denotes the real data. As a constraint, the state u must comply with the governing physics of the problem (flow in porous medium represented through partial differential equations along with appropriate boundary conditions) as represented by the operator g.
This objective function may involve additional terms, such as regularization (e.g. S representing a regularization function that incorporates a-priori information into the objective), or additional constraints (e.g. positivity or bounds for some parameters).
Unfortunately, with respect to ill posedness and uncertainty, the acquired production data do not typically convey sufficient information for a complete and stable recovery of the subsurface properties and, consequently, the resulting solutions are corrupted by the intrusive null space of the solution space. With respect to that concern is that the sensitivity of the acquired data at the wells towards changes in the model parameters away from the wells is negligibly small.
Despite efforts to supplement missing information by means of multi-modality (e.g. incorporation of seismic, electromagnetic (EM), gravity data) inversion, or through incorporation of a-priori information (via regularization, re-parameterization), a great extent of uncertainty in determining the subsurface properties remains. This uncertainty is typically accounted for through extensive sampling of the model prior space, that is, multiple plausible subsurface realizations are generated. These can account for uncertainty in model parameter distribution, in candidate well placement or with any other uncertain parameters.
With respect to the “model prior space” mentioned above, it is noted that as uncertainties are involved in this problem, the problem is dealt with in Bayesian inference settings. In these settings, a goal is to get samples of the posterior probability (model probability distribution given the data). Using Bayes theorem, the posterior distribution is proportional to the product of the likelihood (probability of the data, given the model) by the prior probability (the probability of the model). In other words, this means that the posterior distribution is a compromise between trusting merely the data (likelihood) and trusting merely the prior knowledge regarding the model (the prior probability). The model prior space is a space that includes all model configurations and their assigned probabilities.
Since these realizations are drawn from prior distribution of the model space, it is unlikely that given a set of prescribed controls y, the realizations would conform with the recorded real data d. For that reason, the conventional workflow requires obtaining estimates of the posterior distribution of the uncertain parameters through the aforementioned process of history matching. Once a posterior distribution is obtained, one is typically interested in the way uncertainty is manifested in terms of future forecasts for a given set of future controls. The forecast spread can then be used as a factor for making judicial business and operational decisions.
Of note, as mentioned above, the “posterior distribution” assigns posterior probabilities to each of the possible models. The posterior distribution reflects authentically the settings of the problems, the given data (e.g. production data), and possibly some prior knowledge regarding the model (e.g. a probability may be assigned to each model instance), and a goal is to quantify the probability of having the model in any set configuration.
In the context of large-scale problems and extensive set of realizations, this framework is rendered impractical due to the computationally prohibitive costs of computing multiple history matching processes for each realization.
If not only the prior samples, but also their corresponding posteriors and thereby future forecasts are distinct, there might be no computationally tractable resolution for the problem.
Fortunately, often this is not the case as different model realizations may correspond to (almost) similar dynamic behavior. Given a large set of model realizations, the question that this invention addresses is how to conclude a sub-set that on the one hand captures the dynamic variability of the entire set, yet, is indifferent to dynamic redundancy?